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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 00:01 Prepared using nmeauth.cls [Version: 2000/01/19 v2.0]
Constrained Boundary Recoveryfor Three Dimensional Delaunay Triangulations
Qiang Du1,3 and Desheng Wang 2,3,∗
1 Department of Mathematics, Penn State University, University Park, PA 16802, USA2 Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea,
Singleton Park, Swansea SA2 8PP UK.3 Lab for Scientific and Engineering Computing, Academy of Sciences, Beijing, China
SUMMARY
A new constrained boundary recovery method for three dimensional Delaunay triangulations ispresented. It successfully resolves the difficulties related to the minimal addition of Steiner pointsand their good placement. Application to full mesh generation are discussed and numerical examplesare provided to illustrate the effectiveness of guaranteed recovery procedure. Copyright c© 2004 JohnWiley & Sons, Ltd.
key words: guaranteed constrained boundary recovery, three dimensional Delaunay triangulation,
Steiner point insertion and placement, mesh generation and optimization
1. Introduction
Efficient and robust mesh generation is an essential part of many scientific and engineeringcomputational challenges. For unstructured mesh generation, Delaunay-based algorithms havebecome very popular[1-17]. For such algorithms, the input data are often given in a discreteformulation of the domain boundary which, in the three dimensional space, may represent asurface triangulation of the boundary. Delaunay-based methods usually first produce an initialtriangulation that forms the convex hull of the boundary points which may not match withthe prescribed boundary surface, that is, the triangulation may not satisfy the constraints(edges and faces in 3D) imposed by the surface triangulation. This leads to the problem ofrecovering the boundary geometric constraints from the initially constructed triangulation, orsimply, the problem of boundary recovery. Such problems have been successfully resolved intwo dimensional spaces[3,4], while they are still under active investigation in three dimension.
∗Correspondence to: Civil and Computational Engineering Centre, School of Engineering, University of WalesSwansea, Singleton Park, Swansea SA2 8PP UK.
Contract/grant sponsor: China State Major Basic Research Project G199903280 and US NSF-DMS 0196522,NSF-ITR 0205232
Copyright c© 2004 John Wiley & Sons, Ltd.
2 DU AND WANG
Roughly speaking, there exist two types of three dimensional boundary recovery procedures.The first is the conforming boundary recovery which is done by applying edge/face splittingto recover a constraint as the concatenation of edges/faces. It usually requires the insertionof points to the missing constraints [13-16]. The effectiveness of such recovery methodshas been demonstrated in many cases. Recently, we have also developed a very robust andtheoretically guaranteed scheme for conforming boundary recovery [20]. The second approachis the constrained boundary recovery, which does not allow extra points being added to themissing constraints during the recovery. The constrained boundary recovery respects the localsizing specification of the input boundary triangulation and offers more robustness than theconforming recovery when meshes from two connected regions are merged together. We notethat both the conforming or constrained boundary recovery may be applicable with input beinga surface triangular mesh, however, the resulting tetrahedral mesh after boundary recoverymay not be strictly Delaunay. Due to the Schoenhardt configuration[17], it is well known thatthe success of constrained boundary recovery may rely on the occasional insertion of interiorSteiner points [17].
To list a few well-studied methods for constrained boundary recovery, we begin with thework by George et al.. In [12], they introduced ingenious techniques to reconstruct a prescribedsurface triangulation. The surface edges and faces were recovered by a series of edge/face swaps(or flips), and occasionally some heuristic insertions of interior points (Steiner points). Theheuristic procedures do not work in all cases, as several examples in [11] provides evidencesthat the above methods may fail in some situations. In [13], Weatherill and Hassan introducedmethods in which they first insert intersections points on missing edges and faces to get aconforming recovery, then they try to delete the inserted points through retriangulating thetetrahedra set connected to those points. The existence of Schoenhardt configuration againimplies that the success of such deletion can not always be guaranteed. Until now, there isno published algorithm on constrained boundary recovery that claims theoretically guaranteefor its success in all cases. A number of difficult issues remain to be resolved, such as theplacement of the Steiner points, the minimum bound on such points, etc [17].
In this paper, a new constrained boundary recovery algorithm is presented which combinesconforming boundary recovery with a procedure called added-points-splitting to give aconstrained recovery of the missing boundary constraints. The convergence of the method inall cases is theoretically proved. Our approach is different from the previous works [12-15,17]where one first recovers all the missing edges and then faces. Briefly, in our approach, for eachmissing face, we first recover its missing edges (if there exist) together itself by the modifiededge/face splitting method proposed in [20]. This results in the conforming recovery of themissing face, as some points are added to its missing edges or the interior of the face. Then thepoints added to each missing edge of the missing face are split, one by one, into two interiorSteiner points located away from the missing face. The splitting continues until a completeconstrained recovery of the missing edge. Once all the missing edges of the face are recovered,we again apply the splitting operations sequentially to the added points on the face until therecovery of the missing face. The splitting operation includes a directional perturbation and aconstrained Delaunay insertion of a face-symmetric point. The cavity is appropriately chosenso that a basic flippable local tetrahedra set can be generated. For a missing edge, by flippingthe local tetrahedra set, the edge or a part of it is recovered in a constrained manner; for amissing face with already recovered edges, the number of added points are gradually reduceduntil the complete recovery of the face.
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 3
Note that in the above constrained recovery procedure, when a missing face is recovered,each point added in the initial conforming recovery procedure (if there exists any) is split intotwo interior Steiner points positioned on the two sides of the recovered face. For applications tofull mesh generations, either vertex suppression is performed (if allowed), or the added pointsare relocated to suitable positions with respect to the given sizing field. For practical 3DDelaunay constrained boundary recovery, we first perform several basic swapping operationsdescribed in [20] to recover a large portion of missing constraints. This will involve some simplelocal operations. Then, we apply the proposed recovery method to the remaining missing faces.This combination will decrease the added Steiner points greatly and also leads obviously toconvergence.
Overall, our proposed constrained boundary recovery is proved to work for all casestheoretically in this paper. Moreover, our approach also addresses some key aspects of boundaryrecovery: how to recover the missing constraints, where the added Steiner points should be,and how to minimize the number of added Steiner points. We also demonstrate in this paperthe effectiveness and robustness of the proposed method through various numerical meshingexamples.
The remaining part of the chapter is organized as follows: some basic concepts such asVoronoi tessellation, Delaunay triangulation, conforming and constrained boundary recoveryare briefly described in Section 2. The existing conforming and constrained boundary recoverymethods for 2D and 3D Delaunay triangulation are discussed in section 3. The details ofthe proposed new recovery method are presented in Section 4 and Section 5 for 2D and 3Drespectively. The application of the method to full 3D Delaunay meshing is discussed in Section6, along with various numerical meshing examples and meshing statistics. Finally, conclusionremarks are given in Section 7.
2. Preliminaries: some basic concepts
Given a polyhedra domain Ω in Rd (d=2,or 3) and a set of points S = Ziki=1 in Ω, the Voronoi
region Vi corresponding to the point Zi consists of all the points in Ω which are closest to Zithan to other points Zjj 6=i. With the usual Euclidean metric, the union Viki=1 is called aVoronoi tessellation or a Voronoi diagram of Ω. The geometric dual of a Voronoi tessellation isknown as the Delaunay triangulation, which shares the empty ball property, that is, the circum-sphere of each element does not contain any other vertex. This results in the maximizing theminimal angle property in two dimension and the connection of nearest points property ingeneral dimensions. In practical implementation, the Incremental Insertion method [17] maybe followed to construct the Delaunay triangulation. The Delaunay insertion kernel, includingthe construction of the so-called Base, Cavity and Ball, is used for the addition (or insertion)of Steiner point [17]. For a given point set S, the Delaunay triangulation forms a triangularmesh (tetrahedral mesh in 3D) of its convex hull in general.
There are various ways to generate a triangular mesh for a given domain Ω that is inconformity with a given sizing function, i.e., the edge lengths obeying given specifications.Among them, the popular Delaunay-based methods starts with a discretization of the domainboundary in accordance with the given sizing function. For planar domain, this correspondsto the discretization of a piecewise linear polygon, while for three dimensional cases, thediscretization is formed by a closed surface triangulation. Denote the points (vertices) of the
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discretized boundary by S, the next step is to form the Delaunay triangulation of S using theIncremental insertion method. The resulting triangulation forms an initial mesh of the convexhull of S. Then, field points are generated according to the sizing function and inserted into theinitial mesh or the most recently updated mesh. Finally, optimization is performed to improvethe mesh quality.
It is important to elaborate on the fact that the boundary of the initial mesh often does notmatch the discretization of the domain boundary, see Fig1. In other words, some constraints(edges, faces or surface triangles) of the surface triangulation are missing from the initialmesh. This leads to the boundary integrity problem [3,4,12,13]. The retrieving of the missingboundary constraints can be done through either a priori re-definition of the constraints, if asatisfactory topological and geometrical equivalent can be found in the initial mesh, or by aposteriori recovery of the topological and geometrical requirements of the missing constraint.The latter is called the boundary recovery [3,4,12,13,17].
The a posteriori recovery of a missing constraint from the initial mesh consists of some localoperations to recover the constraint. Based on the characteristics of the local operations, theboundary recovery approaches can be classified into two categories: conforming and constrainedboundary recovery [17]. In conforming boundary recovery, usually additional points are addedto the missing constraint, as shown in Fig1(3); while in constrained boundary recovery, nopoints are allowed to be added to the missing constraint, and only occasional interior Steinerpoints are inserted, as shown in Fig1(4) (the most right).
If the input is a piecewise linear complex(PLC), Schewchuk proposed the constrainedDelaunay triangulation (CDT) for boundary recovery [18]. For a PLC, Schewchuk’s algorithmfirst adds points to missing edges of the PLC in order to reconstruct it as a union of edges; thena CDT is constructed without introducing any interior points. Such a recovery scheme preservessome nice theoretical properties of Delaunay triangulation and works well with provably goodDelaunay refinement methods for tetrahedral mesh generation. However, as extra points onmissing edges are added, such an approach belongs to the conforming boundary recoverycategory.
For practical applications, whenever possible, the constrained boundary recovery is oftena better alternative to the conforming counterpart. In the next section, we will discuss theexisting methods for 2D/3D conforming and constrained boundary recovery.
3. Existing boundary recovery methods for conforming and constrained 2D and 3Dtriangulation
In this section, we briefly recall the features of some typical existing conforming and constrainedboundary recovery approaches. Our focus is on the extendibility of two dimensional methodsto the three dimensional case.
3.1. Conforming boundary recovery
Many existing methods for 2D conforming boundary recovery have one common characteristic:the addition of points on missing edges through the typical Delaunay insertion procedure orothers, see for example [18]. These points can be intersections points of the missing edge withthe available mesh or the midpoints of the missing edge [17]. Due to the nearest connection
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 5
Figure 1. Boundary recovery for 2D case, from left to right: (1) boundary discretization; (2) the missingspecified edges: AC, BD; (3) conforming boundary recovery; (4) constrained boundary recovery.
Figure 2. Conforming recovery for a missing face: (1) a missing face and its two missing edges AC,BC; (2) the face is recovered as a concatenation of sub-faces.
property of the Delaunay triangulation, a missing edge can always be recovered as a union ofedges in the mesh. The difference of the existing 2D methods often lies on the number andthe positions of the added points on the missing edges. Also, when recovering a missing edge,most of the algorithms do not pay any attention to whether an existing or a already recoveredconstrained edge could be destroyed.
Comparing with 2D cases, three dimensional conforming boundary recovery is different inseveral aspects. At first, a missing face may contain one or more missing edges, i.e., whenrecovering the face, its missing edges also need to be recovered. Secondly, one often firstrecovers all the missing edges then proceeds to the missing faces, or reconstructs each face(triangle) together with its missing edges independently. A similarity with the 2D procedureis the addition of intersection points or some other kinds of points (analogous to midpoints foredges) on missing constraints (edges or faces), see a detailed discussions in [17] on the threedimensional conforming boundary recovery. When recovering a missing edge or face, as in thetwo dimensional case, in order to protect the Delaunay property, most existing methods donot protect the existing recovered constraints, that is, no consideration is given to whethersome already-recovered constraints would again be destroyed. This could result in recursivethe addition of points. Special treatments must be required for the convergence. For a missingface, i.e., a surface triangle, the final reconstruction is a concatenation of sub-triangles in themesh after the conforming recovery, see the illustration in Fig2.
In [20], the authors proposed a theoretically guaranteed three dimensional conforming
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boundary recovery which recovers the missing faces one by one. For each missing face, itsmissing edges are recovered first followed by the recovery of the face. The recovery processuses the addition of nearest-mid intersection points for missing edges and the addition ofintersection points for missing faces. The Delaunay insertion kernel is modified so that itprotects any existing constraint or any recovered parts of a missing constraint. This guaranteesthe convergence of a complete conforming boundary recovery. Moreover, when adding pointsto a missing constraint, some neighboring missing constraint may be recovered simultaneously.
In order to achieve constrained boundary recovery, further modification of the above methodis needed. In particular, the recovery of a missing face and its edges should be independentlydone. In essence, such a strategy makes the method work in both two and three spacedimensions.
3.2. Constrained boundary recovery
Among methods for two dimensional constrained boundary recovery [3,4,17], perhaps the mostpopular and the easiest to implement method is the random or serial diagonal edge swappingalgorithm proposed by P.L.George in [13,17]. For a missing edge, the recovery procedure beginswith finding the intersection edges of the missing edge with the existing mesh (actually withthe edges of the mesh), followed by randomly or sequentially swapping the intersection edges,if the operation is allowed. If the edge is recovered, the process is terminated; else, the abovetwo steps are iterated until the final recovery, see the illustration in Fig3. The diagonal edgesswapping is a local operation, hence making the recovery efficient. The recovery of each missingedge is independent from any other missing or constrained edge, and accordingly does not affectthe recovery of other edges. Furthermore, due to the fact that for an arbitrary 2D piecewiselinear polygon, it is always guaranteed that one may triangulate it with its boundary edgesbeing constrained and without introducing any inner field points (Steiner points), we see fromthe above two properties that the two dimensional constrained boundary recovery problem iscompletely resolved by applying the above diagonal edges swapping [17].
For three dimensional constrained boundary recovery, however, the solution is not ideal. Dueto the Schoenhardt configuration shown in Fig4, a constrained triangulation for an arbitrarypolyhedron is not always guaranteed without introducing some inner field points, i.e., Steinerpoints [17]. In [12], P.L.George introduced a method combined with some heuristics that usesa series of complex edges and faces flipping for the recovery, and occasionally some Steinerpoints are added to generate the basic flippable configurations. Though the method has beendemonstrated to be very successful in numerical testing, no theoretical guarantee can beclaimed on the success of the method in all cases [11] as it does not adequately addressissues like where to place the Steiner points and how many Steiner points are needed. Later,in [13], Weatherill and Hassan introduced another method which first generates a conformingboundary triangulation with the insertion of intersection points on the missing faces or edges,then the Ball (the elements emanating from the point) of each added point is re-triangulatedto delete the added point. Since the deletion can not always be done successfully, this methodalso does not offer guaranteed success. Up to now, there has been very few further developmenton three dimensional constrained boundary recovery [17].
Since the two dimensional conforming boundary recovery approaches often can be easilyextended to three dimensional cases, a natural question arises in comparison: can the 2Dconstrained boundary recovery algorithm be extended to 3D cases? To answer such a question
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 7
Figure 3. Diagonal edges swapping for a missing edge.
Figure 4. The Schoenhardt configuration
affirmatively, it is crucial to develop the right methodology for the two dimensional constrainedboundary recovery first. In the 2D case, a series of local swapping may be applied to recoverthe missing edge and there is no need to introduce any Steiner point. In the 3D case, however,the issue of Steiner points starts to surface. This difference tells us that even for the 2D case,we should develop a new method which considers the generation and deletion of the Steinerpoints. In some sense, the diagonal edges swapping method is hiding the problem of Steinerpoints as their deletion can be done successfully in 2D. Thus, we see naturally where theimprovement needs to be made in a new approach: for the recovery of a missing edge, thenew method should include the addition of intersection points, the point splitting, the naturalgeneration of Steiner points, the deletion of the Steiner points and the final recovery of missingedges. The two dimensional version of such an approach is described in the next section whilein Section 5, the method is naturally extended to the three dimensional cases, and it solvesthe three dimensional constrained boundary recovery problem by addressing several important
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Figure 5. Diagonal edge swapping: Swap2-2
issues with satisfaction.
4. A new 2D constrained Delaunay boundary recovery
Before introducing the new algorithm for two dimensional constrained boundary recovery, wefirst present some basic definitions.
DEFINITION 4.1. Let P be a vertex. All the elements (i.e., triangles) emanating fromP is called the Ball of P , denoted by Ball(P). For an interior vertex, its Ball is a closed andconnected polygon in 2D and a polyhedron in 3D.
DEFINITION 4.2. Let 4ABC and4ABD be two triangles sharing the common edgeAB, shown in Fig5. If the quadrilateral ACBD is convex, we can swap the diagonal edge ABto CD, and form two new triangles 4ACD and 4CBD. This swapping is called Swap2-2.
The importance of Swap2-2 is the direct recovery of edge CD in the boundary recovery,which has been discussed before. It also plays a role in preserving the Delaunay property orthe empty disc property and in mesh optimization [17].
DEFINITION 4.3. Let P be an interior vertex and its Ball be denoted by Ball(P). Anoperation that gets a triangulation of Ball(P)’s vertices except P itself, with respect to theboundary edges of Ball(P), is called a Vertex Suppression of P .
In Fig6, it is shown in 2D cases there are two methods for the suppression of P . We canfirst delete P and then re-triangulate the remaining polygon; also, we can perform a series ofSwap2-2 operations to the inner edges of Ball(P) until the existence of a configuration wherethree edges are connected to P , then, we can remove P and get the triangulation we want.
DEFINITION 4.4. Let P and Ball(P) be defined as above. Then, there must be a polylinewhich consists of some inner edges connecting P and divides Ball(P) into two connected andnon-overlapping parts. Such a division is called a Polyline Cutting.
In Fig7, the polyline APB and CPB both form Polyline Cuttings of Ball(P).DEFINITION 4.5. Let P and Ball(P) be defined as above and shown in Fig8(4), where
P is on the line AB. We call the following operations Point Splitting, i.e., the splitting of P ,by mirror symmetry with respect to AB.
(a). First, P is perturbed to P 1 along the direction normal to−−→AB. Denote the symmetric point
(mirror image) of P 1 with respect to the line AB by P 2. Let the perturbation distance of P beε (here, ε is required to be small enough such that both P 2 and P 1 stay inside Ball(P), such
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 9
Figure 6. Vertex Suppression: (1) ball(P); (2) method 1: retriangulation; and method2: from (3) edgesswapping to (4) final vertex deletion until (5) final swapping
Figure 7. Polyline Cutting: (1) choosing the polyline; (2) cutting into two parts
an ε always exists and it it related to the point P and its ball configuration). An example ofthe perturbation is shown in Fig8:(2)-(3).(b). The second step (shown in Fig8) is to add the symmetric point P 2 in the Delaunay mesh.
In the addition, we let AP 1B be a Polyline Cutting of Ball(P) and choose the part whichcontains P 2 as the Cavity, shown in Fig8.(c). Insert P 2 into the mesh using the classical Delaunay insertion kernel[17], with the Cavitychosen as in the above. This operation is shown as in Fig8(5).(d). Perform Swap2-2 to the edge P 1P 2, then AB becomes an edge of the final mesh. The laststep in Fig8(6) gives an illustration of this operation.
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Figure 8. Point Splitting: (1) ball(P); (2) perturbing P to P 1; (3) adding P 2; (4) choosing cavity; (5)mesh updating; (6) final swapping
By the above splitting, P is replaced by two points P 1, P 2, and AB becomes an edge ofthe final triangulation. If AB is an constrained edge or a part of some constrained edge, wehave recovered a missing edge or a part of it through the splitting. Furthermore, two innerfield points, i.e., two Steiner points are generated naturally and they can be subject to vertexsuppression. Obviously, for two dimensional cases, we can delete them through the vertexsuppressions. For 3D, the vertex suppression cannot not always be guaranteed to work, eventhough it can be performed in 2D.
With the above preparation, we now come to the description of our new algorithm forconstrained 2D boundary recovery. For each missing edge, first, we perform the proposednearest-mid intersection points insertion to recover the edge in a conforming manner; then, foreach added point on the edge, we sequentially perform point splitting and vertex suppressionsto recover a part of or the whole missing edge in a constrained manner. This recovery processis shown in Fig9. The pseudo-code form of the algorithm is given as follows:
Algorithm: Point-Splitting Boundary Recovery.
For each missing edge (say AB), perform the operations:(a). Obtain a conforming recovery of AB by adding the nearest-mid intersection points
on AB (see Section2). AB is then reconstructed as a union of edges in the mesh.(b). For each added point (say P) on AB, perform the operations:
(c). Perform Point Splitting to P to obtain two Steiner points P 1, P 2 inthe mesh. A part of AB is then recovered in a constrained form.
(d). Perform Vertex Suppressions to P 1 and P 2.Enddo
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 11
Figure 9. Point-Splitting Constrained recovery for a missing face: (0) edge AB is missing; (1)conforming recovery for AB; (2) splitting of the added point P ; (3) suppression of Steiner points;
(4) splitting of added point Q; (5) final recovery
Enddo
In the above boundary recovery procedure, the convergence of the initial conformingboundary recovery for each missing edge has been theoretically proven in [20]. The splittingof each added point recovers a part of the missing edge (or the whole edge) in the constrainedform. With such splittings, the recovered part of the missing edge becomes longer and longerand after a finite step, the complete constrained recovery is achieved. Furthermore, the vertexsuppressions can always be done successfully. Thus, we have the following theorem for theconvergence of the proposed algorithm:
Theorem 4.1. The above Point-Splitting Boundary Recovery algorithm is convergent, andthere is no Steiner points to be included in the final mesh.
Generally speaking, the above new algorithm is more complex than the diagonal edgesswapping introduced by P.L.George[17] but it captures the essence of constrained boundaryrecovery in having a suitable treatment of Steiner points. Through the insertion of nearest-midintersection points, a conforming recovery of the missing edge is obtained, which provides theinitial positions for the Steiner points introduced later; through Point-Splitting, it naturallygenerates the Steiner points and recovers the constrained edge or a part of it; through Vertex-Suppression, it deletes the generated Steiner points naturally, while preserving the recoverededges. More importantly, it leads naturally to a three dimensional constrained boundaryrecovery method that includes, similar to 2D cases, conforming 3D boundary recovery, vertexsplitting for the natural generation of Steiner points and the recovery of constraints, vertex
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Figure 10. Poly-triangles Cutting.
suppression or optimal placement for the treatment of the Steiner points. Details on theseextensions are given in the next section.
5. 3D constrained Delaunay boundary recovery
In three dimension, the recovery procedure contains edges recovery and faces recovery whichwill be treated separately, even though there are many common features between them.
5.1. Basic definitions and Properties
We again begin with some definitions.
DEFINITION 5.1. In a three dimensional tetrahedral mesh, all the tetrahedra emanatingfrom a vertex is called its Ball and all the elements connecting an edge is called its shell.
DEFINITION 5.2. Let P be an inner vertex of a tetrahedral mesh with ball Ball(P). Anoperation is called a Vertex Suppression of P if a constrained tetrahedralization is obtainedfrom the surface points of Ball(P) with the input being the surface triangles of Ball(P) butwith P not being a vertex of the tetrahedralization.
The suppression of an inner vertex is always guaranteed in the 2d case, but not in the 3Dcase due to the existence of the Schoenhardt configuration.
DEFINITION 5.3. Let P and Ball(P) be defined as above. A subset of the inner trianglesof Ball(P) is called a Poly-Triangle Cutting of Ball(P) if it is connected and divides Ball(P)into two connected and non-overlapping subsets.
As an illustration, the set of inner triangles PQ1Q2, PQ2Q3, PQ3Q4, PQ4Q5, PQ5Q1forms a Poly-Triangle Cutting of Ball(P) in Fig10.
DEFINITION 5.4 Let 4ABC be a triangle with O as the centroid, and P be a pointshown in Fig11. The vertices of 4ABC are called directionally oriented (or ordered), if the
outer normal direction−−→ON is defined as
−−→AB ×−−→BC. The triangle 4ABC is said to be visible
to P , if−−→ON · −−→OP > 0.
The visibility of 4ABC to P is equivalent to the signed volume of the tetrahedron ABCP
being positive, or, P being on the side of 4ABC that−−→ON points to. In Fig11, 4ABC is
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 13
Figure 11. Visibility of a face to a point: (1) directionally oriented; (2) visible to P , not to Q
visible to P , but not to Q. If−−→ON · −−→OP = 0, P is on the plane defined by the three points
A,B,C.The incremental Delaunay insertion kernel for adding a new point (say P ) consists of three
parts: finding the Base, constructing the Cavity, forming the Ball and updating the mesh[17].In practice, due to the round-off errors, the initially formed Cavity is usually processed bya correction procedure to ascertain its validity [17] which is defined as: first, it should beconnected and have no inner vertex; then, each boundary face (i.e. triangle) of the cavityshould be visible to the added point P . Let Tc be the initially defined cavity for P , we nowrecall a simplified version of the correction procedure proposed in [20] in the following pseudo-code form.
Algorithm: Cavity Correction.
1 Find Fimi=1, the external faces F of Tc; define the m simplices Kj : Kj = Fj , Pi+1with Det(Kj) being the signed volume of Kj .
2 Set T = Tc, loop over j = 1 to mIf Det(Kj) > 0, continue;If Det(Kj) < 0, T = T −Kj ;If Det(Kj) = 0, T = T ∪K,
where K is the simplex of Told, with face Fj , not in Tc.3 If T is not affected by step 2, end; else update the set of F and go back to step 2.
DEFINITION 5.6. With reference to Fig12, the three basic single-step swappings are
1 Swap23 : exchange abcd and abce with deab, debc, deca, where de intersects the interiorof abc;
2 Swap32 : the inverse of Swap23 ;3 Swap44 : exchange abde, bcde, abfe, bcdf with acde, abce, acdf, abcf, where ac intersects
the interior of bde or bdf.
Swap23 results in the generation of an edge de and will be used in the recovery of missingedges; Swap32 results in the generation of an face abc and will be used for recovering missingfaces; and Swap44 can be used in both edges and faces recovery. The first two swaps will bethe most basic swapping operations in our 3D constrained boundary recovery.
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Figure 12. Basic swappings
These three basic operations are not enough to recover all missing constraints in practice.In [17], P.L.George proposed a series of complex swapping operations. We describe two morebasic tetrahedra configurations and transformations (or say flippings) here which forms thekey parts of our 3D constrained boundary recovery.
DEFINITION 5.7 Cap-Configuration and Cap-Transformation: With reference toFig13, P,Q are two plane-symmetric points with respect to the plane defined by the co-planarpoints P1(= A), P2, ..., Pn−1, Pn(= B). And AB intersects with the interior of all the facesPQPjn−1
j=2 . The set of tetrahedra PPiPi+1Qn−1i=1 , whose union is called a Cap-Configuration.
In the above tetrahedra union, AB does not exist as an edge. In Fig13, it is shown on the rightthat if we triangulate the 3D-polygon Pjnj=1 into a set of triangles 4in−2
i=1 , we can exchange
the above tetrahedra set with the union of three parts: ABPQ, P4in−2i=1 , 4iQn−2
i=1 . Here,we generate a new edge AB in the exchanged tetrahedra union. Such a exchange is calledCap-Transformation.
Obviously, if AB is a missing constrained edge or a part of a constrained edge, through theCap-Transformation, it can be recovered in a constrained manner. In the simplest case, n = 3,the Cap-Transformation is reduced to Swap23, thus it gives a natural extension of Swap23.
The above co-planarity condition of Pjnj=1 can be relaxed. Actually, we only need thatAB intersects the interior of the faces PQP2, ..., PQPn−1.
In the configuration of Swap32, edge de intersects only one triangle abc. If there are morethan one triangle, we have the following extension of Swap32.
DEFINITION 5.8 Shell-Configuration and Shell-Transformation: With referenceto Fig14, A and B are two points on the two sides of the plane defined by the co-planarpoints Pjnj=1. Let AB intersect the interior of the 3D-polygon P1P2...Pn. With these n+ 2
points, we form a tetrahedra union: U∗ = ∪n−1i=1 PiPi+1BA ∪ PnP1BA, which is called a
Shell-Configuration. As in the Cap-configuration, if the 3D-polygon P1P2...Pn is triangulatedto be a union of triangles 4in−2
i=1 , we can exchange U∗ with the union of 4iAn−2i=1 and
4iBn−2i=1 . Such an exchange is called a Shell-Transformation.
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 15
Figure 13. (1) Cap-configuration; (2) Cap-transformation.
Figure 14. (1) Shell-configuration ; (2) Shell-transformation.
The simplest Shell-Conformation with n = 3 is just the basic operation Swap32.We now present two important points splitting operations for removing edge-added-point
(point added to an edge) and face-added-point (point added to a face) which will results inthe constrained recovery of edges/faces.
Algorithm: Edge-Added-Point Splitting.
Suppose 4ABC is a missing face with edges either missing or not missing and aconforming recovery has been performed through the addition of intersection points sothat4ABC has been reconstructed as a union of triangles in the existing mesh. Withoutloss of generality, we make reference to Fig15:(1), where two points are added on AC,one point is added on BC and one interior point is added on the face ABC. Relabel thepoints set c, g, f, e2 by P1, P2, P3, P4 for convenience, see Fig15:(2). The ”Splitting”of the added point e1 means that P1P4 is generated as an edge in the updated meshafter the ”Splitting”, i.e., a part of the edge AC is recovered in a constrained manner.Such an operation for e1 is called an Edge-Added-Point Splitting. In more details, theoperation is as follows:
(a) First, we perturb e1 in the direction−→N1 (
−→N1 =
−−→AB × −−→BP1) by a distance ε1, i.e.,
e1 = e1 + ε1 ·−→N1 (see Fig15:(3) for an illustration of its new position).
Then, we perturb e1 in the direction−→N2 by a distance of ε2. Here,
−→N2 is parallel to the
plane defined by ABP1 and is vertical to−−→AP1. Fig15:(4) shows the newly perturbed
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16 DU AND WANG
position of e1. Denote the symmetric point of e1 (with respect to the plane ABP1) by e∗1.The two distances ε1, ε2 are chosen to be suitably small such that e1, e
∗1 are both contained
in the ball of e1 and all the faces e1PjPj+1n−1j=1 are visible to e∗1 (such distances ε1, ε2
can obviously be defined in all cases and they are related to the added point and its ballconfiguration).
(b) Insert e∗1 into the existing mesh using the constrained Delaunay insertion kernel :First, the cavity of e∗1 (Denoted by Tc) is initialized to be the ball of e1, i.e., Tc = Ball(e1).Then, Tc = Tc − ti, where ti include the tetrahedra having e1PiPi+1, i = 1, 2, 3as faces and lying ”above” the triangle 4ABP1 (i.e. in the same direction as
−→N1). With
reference to Fig15:(5), ti are the tetrahedra e1P3P4f1, e1P2P3f2, e1P1P2f2. which donot intersect the triangle 4ABP1. Obviously, Tc may not be a valid cavity.Then, the third step is to perform the Cavity Correction procedure described above toTc. Denote still the updated and valid cavity by Tc. Since the faces e1PiPi+1n−1
i=1 arevisible to e∗1, they will be in Tc as boundary faces.Finally, we insert e∗1 into the mesh with Tc being the cavity. According to the constrainedDelaunay insertion kernel, new tetrahedra will be formed by e∗1 and the boundary facesof Tc. Hence, a Cap-Configuration Cc = e∗i f i, f i = eiPiPi+1n−1
i=1 will be generated (seeFig15:(6)).
(c) Perform Cap-Transformation to the Cap-Configuration Cc. Then, P1P4 becomes an edgein the the updated mesh (the final configuration is shown in Fig15:(7)).
The above point splitting procedure deletes an added point e1 on AC and generates twoSteiner points e1, e
∗1, which are on the two side of 4ABC. And a part of the edge AC is
recovered in a constrained manner. Having a suitable definition for the cavity Tc is the key toreach the above result. In conclusion, we have the following lemma for the Edge-Added-PointSplitting.
Lemma 1. The above Edge-Added-Point Splitting operation deletes one added point on aconformly recovered edge once a time, and recovers a part of the edge in constrained form (ifthere is only one added point, this operation results in the complete constrained recovery ofthe edge). Moreover, each splitting generates two Steiner points, and this splitting will neverresults in new added points in the interior of the missing face. Also, it will not affect theexistence of any recovered or constrained face or edge.
We now deal with the splitting of points added to faces.
Algorithm: Face-Added-Point Splitting.
Let4ABC be a missing face after a conforming recovery with its three edges all recoveredthrough the above Edge-Added-Point Splitting operations. Assume that 4ABC stilldoes not exist in the mesh as a face, that is, there are still points added in the interiorof 4ABC. Without loss of generality, suppose three points P1, P2, P3 are added on4ABC (see Fig16:(1) for reference). We call the following operation a Face-Added-PointSplitting which will ”Split” an added point (say P1) into two Steiner points. Denote thetriangles in 4ABC that connect P1 by fi (see in Fig16:(2)), which forms a subpart of4ABC. Now, we describe the Face-Added-Point Splitting operation for P1.
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 17
Figure 15. Operation for edge-added-point-splitting: (1) points added for conforming recovery; (2)re-labeling the added points around e1; (3) first perturbation of e1 in the normal direction; (4)second perturbation; (5) cavity modification; (6) inserting the symmetric point e∗1 and obtain a Cap-
configuration; (7) Cap-transformation.
(a) Perturb P1 in the direction−→N by a distance ε. Here,
−→N =
−−→AB × −−→BC , i.e., the normal
direction of 4ABC. Denote the symmetric point of P1 (with respect to the plane ABC)by P ∗1 . The distance ε is chosen appropriately or small enough such that P1, P
∗1 are both
contained in the ball of P1 and the faces fi after the perturbation becomes visible toP ∗1 . (The perturbed configuration is shown in Fig16:(3).)
(b) Insert P ∗1 into the existing mesh via the constrained Delaunay insertion kernel:The cavity for P ∗1 is chosen as follows:Since fi forms a Poly-triangle Cutting of the Ball of P1, Ball(P1)is divided into twoparts. We choose the part which contains P ∗1 as the cavity (denoted by Tc) for ouruse. Since fi are Tc’s boundary faces and are visible to P ∗1 , Tc is a valid cavity for P ∗1 .Inserting P ∗1 into the existing mesh, a Ball-Configuration is generated (see the illustrationin Fig16:(4)).
(c) Perform Ball-Transformation to the generated Ball-Configuration.Two Steiner points P1, P
∗1 are generated and the subpart shown in Fig16:(2) is reduced
to that shown in Fig16:(6), which is equivalent to say that the configuration of theconforming recovery for 4ABC is reduced. The union of new triangles is shown in
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18 DU AND WANG
Fig16:(6), where the added points are reduced by one. In the simplest case where there isonly one added point in the initial union, the result of the transformation gives a completerecovery of 4ABC (consequence of a simplified Ball-Transformation - Swap32 ).
Figure 16. Operation for Face-added-point-splitting: (1) Conforming recovery; (2) A subpart:trianglesconnecting P1; (3) Perturbing P1; (4) Adding the symmetric point P ∗1 and generate a Ball-
configuration; (5) Ball-Transformation; (6) New subpart; (7). Reduced configuration
We have the following lemma for the function and property of the Face-Added-Point Splitting.
Lemma 2. For a conformly recovered missing face with its three edges being recovered inconstrained manner, if we perform the above Face-Added-Point Splitting, an added point onthe interior of the face will be deleted; two Steiner points will be generated, and the face is stillformed as a union of triangles but the configuration for the conforming recovery of the face (ormore precisely, the number of triangles in the union) will be reduced in a strictly monotonicmanner. In the simplest case that only one added point exists, the face is recovered completelyin constrained form. Furthermore, this splitting operation will never influence the existence ofany other existing/recovered constraint.
The 3D constrained boundary recovery procedure is ready to be presented which recovers themissing faces one by one, with the missing edges of each face being recovered simultaneously.Such a procedure is different from previous methods that first recover missing edges, thenmissing faces.
5.2. Conforming Face/Edges Recovery
For each missing face, we apply the addition of intersection points procedure discussed inSection 2, see [20] for details. Here, we simply recall the basic steps.
When adding intersection points, the Delaunay insertion kernel is modified such that the
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 19
insertion of each point will not delete any other existing/recovered constrained edge/face, nora recovered part of a missing constraint. This is realized through the modification to the initialchosen cavity for the insertion point[20]. The non-interference to other constraints results in amonotonic decrease in the number of missing constraints. In [20], when we recover one missingface, its neighboring faces may also be recovered as a by-product. Here, such a by-product isgiven up since the Edge-added-point Splitting obviously will destroy the conforming recoveryof some neighboring faces. However, there are two properties shared by the above conformingrecovery: one is that when a missing edge of a missing face is recovered as a union of sub-edges,then, in the recovery of the connecting faces of this edge, the recovery of this edge is preserved,i.e., the common edges are already recovered. This will keep the conformity of the recovery ofthe missing faces. Another important property is that when all three edges of a face have beenrecovered but the face still missing, the subsequent addition of the interior intersection pointsis independent from the addition operation of intersection points for other missing faces. Thiswill guarantee the convergence of the recovery.
5.3. Constrained Edge Recovery
Let f be a missing face and has been recovered in a conforming manner via the additionof intersection points described earlier and in [20]. If it contains edges on which intersectionpoints have been inserted, then the first step for the constrained recovery of face is to performthe Edge-Added-Points Splittings for the added points on these edges. The pseudo-code formfor the algorithm is given as follows:
Algorithm: Constrained Edge Recovery.For the three edges (say ei, i = 1, 2, 3) of f , do the following operations:
if there are points Pij , j = 1, ..., n added on ei, thenfor each Pij , perform Edge-Added-Point Splitting.
EndifEnddo
For the above constrained edge recovery procedure, we have the following theorem:
Theorem 5.1. Suppose f is a missing face which has been recovered in a conformingmanner. If it is processed in the above constrained edge recovery procedure, then its threeedges can be recovered in constrained manner through a finite number of Edge-Added-PointSplittings.
Proof. For an edge with intersection points being added on, by Lemma 1, we have thateach Edge-Added-Point Splitting recovers a part of the edge in constrained manner. And thenumber of added points is monotonically decreasing. Since there are only finitely many addedpoints, a configuration with only one added point will be reached. From Lemma1, the finalEdge-Added-Point Splitting for this last added point results in the complete recovery of theedge. Furthermore, as each Edge-Added Point Splitting is performed in the Ball of the addedpoint, it is independent of the recovery of other edges. Moreover, each Splitting will not resultin new points being added. Hence, the number of required splittings is just equal to the numberof initially added points which is finite.
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20 DU AND WANG
5.4. Constrained Face Recovery
Suppose f is defined as in Section5.3 and its three edges have been recovered by the aboveconstrained edge recovery procedure. If f is still formed as a union of triangles, i.e., there arestill added points in the interior of the face, then, we perform the Face-Added-Point Splittingfor its added points for the final constrained recovery of f . The pseudo-code form for suchrecovery procedure is simply given as:
Algorithm: Constrained Face RecoveryFor the interior added points Pi, i = 1, ..., n of f , do:
Perform Face-Added Point Splitting for Pi.Enddo
We now give the following theorem:
Theorem 5.2. Let f be defined as in Theorem 5.1 but with its three edges all recovered byEdge-Added-Point Splitting. If there are still interior points added on f , one can perform theabove constrained Face Recovery operation for f to recover f in constrained manner througha finite number of Face-Added-Point Splittings. Moreover, its recovery will not affect otherexisting or recovered constrained faces.
Proof. The proof is straightforward. We know from Lemma2 that each Face-Added-PointSplitting reduces an added point and reduces the configuration for the conforming recovery ofthe face in a strictly monotonic manner. As the number of added points is finite, eventually,we get to the situation that there is only one added point on f . Then, the final Face-Added-Point Splitting for this point is reduced to the simple operation Swap32 and it results in theconstrained recovery of f . Furthermore, by Lemma 2, each Face-Added-Point splitting will notgenerate new added point, hence, the complete recovery needs only finite splittings, or moreprecisely, the number of the splittings is equal to the number of points added in the interiorof the face f .
Combining Theorem 5.1 with Theorem 5.2, we have the following conclusion:
Theorem 5.3. Let f be a missing face in a 3D Delaunay tetrahedralization. By performingthe above described three operations: conforming recovery through adding intersection points,constrained edge recovery through Edge-added-Point Splitting, and constrained face recoverythrough Face-Added-Point Splitting, we can recover f in constrained manner through finitesteps of operations. And the constrained recovery of each missing face in the initial tetrahedralmesh will not delete any existing/ recovered face.
In conclusion, for an initial Delaunay tetrahedral mesh, if we perform the recovery procedurefor its missing faces, we can get a constrained 3D Delaunay tetrahedralization in all cases. Ofcourse, the above recovered tetrahedral mesh may not be strictly Delaunay [17].
In practice, a large portion of missing faces/edges can be recovered through the three basicswappings: Swap23, Swap32, and Swap44. These operations are cost-effective. Hence, like in[20], we can first perform these operations for some missing constraints; then, for the remainingmissing ones, we perform the above constrained recovery procedure. The pseudo-code form for
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 21
this combined constrained 3D boundary recovery scheme is as follows:
Algorithm: Combined 3D Constrained Boundary Recovery.
(a) For each missing edge or face, compare its configuration with those of Swap23, Swap32 orSwap44. If a match is found, perform the corresponding swapping to recover the missingconstraint.
(b) For the remaining missing facesfi, i = 1, ..., n do:Perform conforming boundary recovery for fi by adding intersection points.Perform Constrained Edge Recovery for fi’s edges if there are added points on them.Perform Constrained Face Recovery for fi if it is still not recovered in constrained
form.Enddo.
Using the above constrained boundary recovery, we can derive successfully a constrained 3DDelaunay (not strictly) tetrahedral mesh from an initially given Delaunay tetrahedral meshwith a theoretical guarantee in all cases.
This above procedure addresses several important aspects of the open problems related tothe 3D constrained boundary recovery, such as where to place Steiner points, when to generateSteiner points, how many Steiner points are needed. There are naturally other issues whenone considers the full mesh generation process. Those issues are discussed further in the nextsection.
6. Application to 3D full Delaunay mesh generation
When applying the above proposed 3D constrained boundary recovery to full Delaunay meshgeneration, the naturally generated Steiner points may result in low-quality elements nearthe missing constraint. To overcome this problem, we propose an approach that combinesthe minimization of the number of Steiner points and an advancing front repositioning of theremaining Steiner points. Let F be a missing face, and Pi be some Steiner points generatedafter the constrained recovery.
If F is a boundary face, one half of these Steiner points are located outside of the domainwhich require no special treatment as they will not participate in the ensuing interior fieldpoints insertion for full mesh generation.
If F is an inner constrained face, all of these Steiner points need to be processed. The firstoperation to be applied on these points is to try to delete them while keeping the recoveredface F protected. Such a deletion procedure goes as follows: when each Edge/Face-added-points Splitting is done, we find the ball Ball(Pi) (or balls) of the generated Steiner point Pi.Delete the point Pi from the ball, and try to tetrahedralize, for instance, via the Advancingfront method, the remaining polyhedron formed by the boundary triangles of the ball withoutallowing the insertion of any additional points. This tetrahedralization can also be called aVertex Suppression to Pi. Obviously, this suppression can not be done for all cases [17]. If thesuppression fails, we simply keep the Steiner point unchanged. The first step described here isreferred as the Steiner points minimization.
Since the objective of processing the Steiner points is to lessen the deterioration in meshquality, naturally, the second operations on the Steiner points are to reposition them to better
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22 DU AND WANG
locations. Once F is recovered, we move the remaining Steiner points to new positions viathe Advancing front points placement method so that the placement is in line with the givingsizing field. For the case of a single Steiner point with respect to F , the operation can beeasily performed. The Steiner point is repositioned to an optimal location so as to generate anoptimal element. For the case of multiple points, more steps need to be taken. One approachis to perform local constrained Laplacian smoothing to the inserted Steiner points. During thesmoothing, the points are always constrained in their Balls [17] in order to keep the validityof the triangulation. If the points can be moved considerably from the initial positions, theinner field refinement (see the later meshing scheme) will inevitably improve the quality of thesurrounding elements connecting them. On the other hand, if the freedom for moving themis limited, such as in a small cavity, it is obviously more difficult to improve the situation.Moreover, the repositioning of the Steiner points can also be performed in the final meshoptimization by mesh smoothing or by the recently developed CVT methodology [19].
To apply the above constrained boundary recovery together with the post-processing ofSteiner points to full 3D Delaunay mesh generation, we again assume that the input is asurface triangulation of the domain boundary, and a sizing field is given or can be derivedfrom the surface triangulation by interpolation. Naturally, the surface mesh should conformwith the sizing field. A complete meshing scheme goes as follows.
Full 3D Delaunay Mesh Generation Scheme:
(a) Input the surface triangulation.(b) Perform the initial Delaunay tetrahedralization in the classical Delaunay insertion kernel
using points in the surface triangulations and some other specified points if they exist.(c) Perform the proposed 3D constrained Boundary recovery to the initial unconstrained
tetrahedral mesh, together with the Steiner points post-processing.(d) Generate inner field points by coupling AFT method with the constrained Delaunay
insertion procedure, insert them into the existing mesh iteratively until the sizing fieldis satisfied[10].
(e) Optimize the mesh with local or global geometrical and topological operations, or usingCVT optimization[19].
(f) Output the mesh.
The above scheme is applied to mesh various complex geometric models. In the following, wepresent four meshing examples. The four geometric models are: a box containing nine stiffness(spheres) which is used for composite material multi-scale simulation; a complex geometrywith many intersecting spheres, which is for simple molecular surface modeling [22]; a complexgeometry with several small cavities and large sizing variation; and lastly the Gulf2 airplanemodel for external flow simulation. The examples includes pictures of surface triangulation,cutting views of constrained boundary and the full 3D triangulation. They are presented inFig17-21. Also, two tables are given to present the recovery data and statistics on the elementquality.
In Table1, the geometric data for each example are provided which include the total numberof vertices in the triangulation, the vertices of the surface triangulation (i.e., boundary vertices),the number of missing boundary faces in the initial tetrahedralization, and the numberof recovered missing faces through the simple swappings (abbr.SW), i.e., through the firststep. We also report the number of intersection points added in the conforming boundary
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 23
Figure 17. A box included with nine stiff inclusions (spheres): (1) the surface triangulations of theouter box and the inner spheres; (2) cutting views of the constrained boundary tetrahedral mesh; and
(3) the final full 3D tetrahedral mesh.
recovery (CBR), the number of Steiner points generated inside the domain and remained aftersuppression. Moreover, the table also contains the ratios (in percentage) of the CPU timing ofthe boundary recovery step to the whole meshing process of various methods.
For the four examples, a large portion (almost 70 percent) of missing faces are recovered bythe three basic swapping operations, which demonstrates that the first step in our combinedconstrained boundary recovery is very effective and worthwhile. For the remaining missingfaces, in example1, 2 points are added for the conforming boundary recovery of 4 missingfaces, and then the points splitting generates 4 Steiner points. And we find these missing facesall appear on the inner sphere surface, hence, these 4 Steiner points are all inside the domain,as we want to mesh both sides of the inner sphere surface. Clearly, we find that vertices ingeneral cannot be suppressed through re-tetrahedralization. Nevertheless, with pure luck, wefind they are all moved to optimal positions with the AFT technique, which will be desirablefor the full high-quality mesh generation. In example2, there are 9 points being inserted for theconforming recovery of missing faces. After the recovery, the same number of Steiner pointsare generated inside the domain, and another 9 Steiner points are outside the domain, whichwill not contribute to the subsequent interior refinement. Also, 4 Steiner points are deleted by
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24 DU AND WANG
Figure 18. An intersection of spheres for simple molecular surface modeling: (1) the surfacetriangulation; (2) cutting views of the constrained boundary tetrahedral mesh; and (3) the final full
3D tetrahedral mesh.
suppression and the remaining 5 points are all repositioned to good locations through AFTplacement. In these two examples, after careful examination, we find that for each missingface, one only needs to generate one or two (on either side of the face) Steiner points. Thecase that more than two Steiner points are needed has never occurred in our numerical tests,which is largely due to the smooth sizing field and a high-quality surface triangulation. This isin sharp contrast with the example 3 where 63 Steiner points are generated in the domain andin the end, 38 of these points still remain inside. We find that there are two cases for whichtwo Steiner points are generated for each missing face, and only one point has been processedwith the AFT technique while a small perturbation is performed to reposition the other point.
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 25
Figure 19. A complex geometry with several small cavities: (1) the surface triangulation; (2) cuttingviews of the constrained boundary tetrahedral mesh; and (3) the final full 3D tetrahedral mesh.
These cases happen near the part of the surface where there is a large sizing variation. But,after the final CVT optimization [19], these points were repositioned to much better locations.The final example, the Gulf2 model has a larger surface triangulation which has 31633 facets,and there are 620 faces missing after the initial Delaunay tetrahedralization. This is largelydue to the existence of a lot of thin triangles in the input surface triangulation. With ourrecovery technique, 184 Steiner points are added inside the domain and 72 of them remainafter the minimization. However, as the model is used for the external flow simulation, thereis substantial freedom in moving these points around. With the AFT technique, they are allrepositioned to optimal places.
We see from the last line of the Table1 that the CPU time ratio of our boundary recoverytakes up only a very small fraction of the total meshing time, and its effectiveness is evident.
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26 DU AND WANG
Figure 20. The Gulf2 airplane modeling: the surface triangulation
Even though we make no comparison with other techniques [10, 17] here, the efficiency is clearlysatisfactory for real applications. All of the above analysis and simulation results support ourtheoretical prediction that the proposed boundary recovery provides a good and efficient 3Dconstrained boundary recovery scheme and a full 3D high-quality meshing.
Table I. Recovery statistics of the examples.
Examples Example1 Example2 Example3 Example4Total vertices 4856 12545 9466 276653Boundary vertices 1466 3442 3107 31633Original missing faces 14 64 225 620Recovered faces through SW 10 50 142 408Points added in CBR 2 9 63 184Steiner pts generated inside domain 4 9 63 184Steiner pts remained inside domain 4 5 38 72CPU ratio of BR/meshing(%) 2.4 3.8 6.9 1.6
In Table2, some statistics of elements quality (after optimization through the constructionof CVDT [19]) measured by the method of [19] are provided. They include the average quality,minimum quality, ratio of good elements, ratio of bad elements, and the number of bad elementsconnecting the added Steiner points. Here, the quality of each element [19] ranges from 0 to 1.0,with 1.0 being optimal. And we say an tetrahedron is a good element if its quality is above 0.4,
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GUARANTEED CONSTRAINED BOUNDARY RECOVERY 27
Figure 21. The cutting view of several parts of the final full 3D tetrahedral mesh of the Gulf2 model.
and a bad element if less than 0.1. The data of the first two examples are all very satisfactory.In the second example, there are several bad elements close to the surface boundary. For thethird example, due to the large sizing variation, the average quality is only 0.702 even afterCVT optimization. And a portion of the bad elements appear near the boundary. The Gulf2model also has a fraction of bad elements near the input sliver triangles. However, in all of thefour examples, none of the bad element is connected with the generated Steiner points. Thisis because we have either repositioned them to optimal locations through AFT techniques orthrough the final CVT optimization. These data demonstrate that our treatment of Steinerpoints is adequate and practical for the full three dimensional mesh generation.
Table II. Elements quality statistics of the three examples
Examples Example1 Example2 Example3 Example4average quality 0.773 0.771 0.702 0.766minimum quality 0.124 0.073 0.041 0.032percentage of good elements 98.2 93.4 90.5 96.8percentage of bad elements 0.0 0.01 0.12 0.005bad elements connecting SP 0 0 0 0
Based on our numerical examples, we also see that in order to obtain a high-quality 3D mesh,the sizing field of the surface triangulation should avoid large variation. This is achievable
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28 DU AND WANG
through surface remeshing which is an active current research field.From the recovery data obtained for the examples, another observation can be made: better
geometric and curvature control in the surface triangulation results in smaller number ofmissing faces, and also better positions for the added Steiner points. Hence, for efficientboundary recovery, not only the sizing variation of the surface triangulation matters, thecurvature variation also influences the resulting constrained boundary recovery. Take thesecond example, the variation of the sizing field is not large, however, the ratio of missingfaces is not small. With some detailed checking, we find that these missing faces are almostall near the parts of the surface where spheres intersect, corresponding to where the curvatureundergoes large changes. Had the surface mesh been curvature-adapted, that is, larger densitywhen curvature is large, the number of missing faces can be reduced, and complexity intreating the Steiner points also goes down, e.g., suffice in using the AFT techniques. Thesame observation is also confirmed by the third example. Future investigations along theselines will be pursued.
7. Conclusion and future work
A theoretically guaranteed algorithm for 3D constrained boundary recovery is presented,which combines conforming boundary recovery with Edge/Face-Added-Points Splitting. In theSplitting of added points, the Cavity is appropriately chosen such that some basic flippableconfiguration are generated, and the corresponding transformation of these configurationresults in a partial or complete recovery of the constraint. The proposed recovery scheme solvesthe open problem of the 3D constrained boundary recovery by providing answers to severalkey issues: successful recovery for all cases without any heuristics, natural generation andplacement of Steiner points through points splitting, minimizing the number of Steiner pointsthrough vertex suppression, and optimizing Steiner points through AFT points placement orother smoothing techniques. Various meshing examples and their geometrical and elementsquality data demonstrate the effectiveness and robustness of the presented boundary recoveryalgorithm and its application to full mesh generation. Future directions of research includethe further optimization of Steiner points, the extension to higher dimensional cases, and theapplication to 3D anisotropic meshing. As suggested in [21], we will also consider the addingAFT-related recovery to the initial simple swapping operations.
Acknowledgment
The authors are grateful to the anonymous referees and Prof.Oubay Hassan of Universityof Wales Swansea for the valuable suggestions and comments that helped us to improve thepresentation of this manuscript.
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